Understanding maths, or writing down the answers

Bipods have two legs, and tripods have three.   At least two of each land on earth, and they have 23 legs.  How many of each are there?

Well, I thought of solving it with equations.  But there are two unknowns and only one equation.  So I borrowed two primary maths books from the excellent Educational Resource Collection (for trainee teachers - hope they also got the ones they wanted!) at work.   Starting with a guess, was the suggestion.  So we did, and with a bit of use of the 2 and 3 times tables, we arrived at an answer.

There is more than one answer, so we tried to find another.   But we did not understand how we had got to the first answer, so we could go not go any further.

I was not sure what to do next.  I am a Primary PGCE drop out.   So, I recommended we (well, my son) ask his teacher.

The next night, my son wrote down his own fraction problems.  He wanted to do them himself, and take them to school to show his teacher.  An admirable application of what he had learned. 

Some were easily solved using the techniques he had learned at school.

One was 5/10 of 48.  He drew 10 boxes and proceeded to share 48 dots between them.  He ended up with some boxes with 6 and some with 5 dots and was not sure what to do next.   I asked how else you could write 5/10?    1/2, he said.   He tried again, with two boxes, and worked it out.  

Then the next was 6/9 of 50.  The same problem with the boxes, and with no solution in sight, he was very upset that he would have to take the answers to school with blank spaces in.  One of his friends had written out some questions and had no blank spaces, he said.   Maybe, I said, he had not come up with any questions that he had not done at school.  Remember, these were questions my son had written himself, and however many times I told him not to worry, because he had come up against some maths he did not know, he was most upset.   Having the answers was more important than knowing how you got there, and more important than saying "I don't know" and realising that the methods you know don't work, even if you don't know why.

So, when devising assessments or assignments, I need to remember - make them so people have to understand a process, as well as having an answer.

Finding your way

Driving somewhere?  Satnav or map reading, that is the question.

Satnav, pluses and minuses:

• - Risk that you will just follow instructions (copy what you are told);
• - You may have no idea how the route is generated;
• - You may not not know where you are in relation to your destination;
• + You don't need to make a choice over route;
• + If you go wrong, the satnav will work out what to do;
• + You can drive solo with a satnav;
• + It is easy to get started with a satnav;
•  - But better if you take time to work out how it functions.

Maps, pluses and minuses:

• + You know where you are in relation to your destination;
•  +/- You have to work out how to get there
• +/- You need to decide which route, if there is a choice to be made;
• - The map won't tell you where the potential bottlenecks are, although it will tell you which is the most major road if you know the symbol;
• - You can't read a map and drive at the same time;
• - If you go wrong, you need to work out yourself what to do.

Our family car recently gained a satnav, a birthday present for one of its drivers.  We have named it Sally Satnav, as it has a female voice. It (she?) has navigated one or other of us around areas we did not know, and worried for us about the correct junction to take off the motorway and what to do next.  And when we missed a turning, Sally recalculated for us.  Our youngest is amused when Sally does not know the name of a road and tells us to "turn left on road", rather than something like "turn left on Leppings Lane".  And we know what time we are likely to arrive.

I remember testing Sally Satnav on a familiar route in the town where we lived at the time.  She wanted to go a different way to the way I had always gone.  It would have worked.

We have had to learn about how Sally works in order to get the most benefit.  How much notice do you get of junctions?  How do you know if that side road is the one?  Can she tell you what lane to be in at large junctions?  Does she know about roadworks or traffic jams?

But, we have had left turns that are almost too sharp to get round, and more than one road that is only just wider than the car.  And cases where the lane markings and the signs at a roundabout say to go one way, and Sally says to go a different way.  Was it a shortcut?   These things, I imagine, are because of the way Sally calculates your route. 

I am much happier with a route if I have done it before, and not that confident at map reading, so a satnav seems ideal for me.  But to find my way around Sheffield, I have decided I prefer maps.  I want to know where I am in the city (north, south, east or west) and if I am near an area I know, and those things are easier with a map.  I begin to build up a picture of the city in my head.   And on longer trips to unfamiliar places with the satnav, I have not enjoyed the feeling of not knowing where I am on a map, not knowing where I am in relation to where I want to be.

Recently, an academic colleague used the image of satnav and map in discussions about levels of understanding of our medical students.   I have been unpacking the image since.

They learn something and can repeat it in an exam, and perhaps explain it to someone, but if they have to apply that thing in a new context, they are less sure.

With a satnav, I know when I have arrived, but don't always understand how I got there.  I am just copying what Sally Satnav is telling me to do.  Some parts of the route might have been difficult (narrow, sharp turns) and I may not know why.   If I go wrong, I can follow new directions, but I don't have to work them out. With a map, I understand how I got there, and where I went on the way, and I also know how near I am to the destination. If I go wrong, it is easier to see where and how it happened, and I have to work out what to do about it.  Both satnav and map require learning - how does Sally Satnav choose a route? - but you can use a satnav without needing to know that sort of thing, whereas maps are not a lot of good if you don't know how to read them and don't study them in advance.

So, I can follow the images in Netanatomy and know the anatomy of the collarbone.  I can answer a question about the anatomy of the collarbone, and tell you about the anatomy of the collarbone. But faced with a patient who has broken their collarbone, how am I going to use that knowledge to help it mend?   I might learn about the mechanism of fracture healing from a YouTube video, but can I use that basic science to advise the patient what they can do while the bone is mending?   And what if it doesn't mend, or they decide to ignore my advice?   How does the science help me decide what to do next?

Final thought in this ramble.  If I know the route, or have done it before, how about using the satnav as a way to follow it?  If I know I apply that basic science, watching the video or looking at the image may enable me to check my basic knowledge before proceeding.

Research Council funding

Two things that I noticed on the same day:

1 - an interview on the BBC Today programme with the head of the STFC, about their recent funding awards - money for large pan-European projects, apparently, and little for basic research in universities. There is an audio piece from a Today journalist here.

2 - an editorial in Nature about the drive to fund projects that involve industry, at the expense of basic research.

The first made my ears prick up (while I was growing up, my father taught Physics). The second caught my eye (two facial metaphors for the price of one) - there is a place, of course, for collaboration with industry, but I do think industry and business should not determine what happens in education, and overemphasis (note my emphasis) on collaboration of this sort could do that. Of course, higher education can find application through industry, but education is not only vocational.